1. **State the problem:** Find the inverse of matrix $$A = \begin{bmatrix}0 & -1 & 1 \\ 1 & 2 & 0 \\ -1 & 0 & -1\end{bmatrix}$$. 2. **Method:** Use row operations on the augmented matrix $$[A | I]$$ to transform $$A$$ into the identity matrix $$I$$, simultaneously transforming $$I$$ into $$A^{-1}$$. 3. **Initial augmented matrix:** $$\left[\begin{array}{ccc|ccc}0 & -1 & 1 & 1 & 0 & 0 \\ 1 & 2 & 0 & 0 & 1 & 0 \\ -1 & 0 & -1 & 0 & 0 & 1\end{array}\right]$$ 4. **Step 1:** Swap row 1 and row 2 to get a leading 1 in the first row: $$\left[\begin{array}{ccc|ccc}1 & 2 & 0 & 0 & 1 & 0 \\ 0 & -1 & 1 & 1 & 0 & 0 \\ -1 & 0 & -1 & 0 & 0 & 1\end{array}\right]$$ 5. **Step 2:** Add row 1 to row 3: $$\left[\begin{array}{ccc|ccc}1 & 2 & 0 & 0 & 1 & 0 \\ 0 & -1 & 1 & 1 & 0 & 0 \\ 0 & 2 & -1 & 0 & 1 & 1\end{array}\right]$$ 6. **Step 3:** Add row 3 to row 2: $$\left[\begin{array}{ccc|ccc}1 & 2 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 1 & 1 \\ 0 & 2 & -1 & 0 & 1 & 1\end{array}\right]$$ 7. **Step 4:** Subtract 2 times row 2 from row 1: $$\left[\begin{array}{ccc|ccc}1 & 0 & 0 & -2 & -1 & -2 \\ 0 & 1 & 0 & 1 & 1 & 1 \\ 0 & 2 & -1 & 0 & 1 & 1\end{array}\right]$$ 8. **Step 5:** Subtract 2 times row 2 from row 3: $$\left[\begin{array}{ccc|ccc}1 & 0 & 0 & -2 & -1 & -2 \\ 0 & 1 & 0 & 1 & 1 & 1 \\ 0 & 0 & -1 & -2 & -1 & -1\end{array}\right]$$ 9. **Step 6:** Multiply row 3 by $$-1$$: $$\left[\begin{array}{ccc|ccc}1 & 0 & 0 & -2 & -1 & -2 \\ 0 & 1 & 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 2 & 1 & 1\end{array}\right]$$ 10. **Final inverse matrix:** The right half is $$A^{-1}$$: $$A^{-1} = \begin{bmatrix}-2 & -1 & -2 \\ 1 & 1 & 1 \\ 2 & 1 & 1\end{bmatrix}$$ **Note:** The user’s final matrix differs in signs; the above steps correct the signs for the inverse. **Summary:** The inverse matrix is $$A^{-1} = \begin{bmatrix}-2 & -1 & -2 \\ 1 & 1 & 1 \\ 2 & 1 & 1\end{bmatrix}$$